Chariho Regional School District Mathematics Curriculum

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  count and use whole numbers
  
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  use models to show quantity
  
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  use one-to-one correspondence in counting objects
  
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  use mathematical language to begin to compare quantities
  
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  associate a quantity with names and symbols for numbers
  
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  Compare whole numbers to each others and to benchmark numbers 5,10
  

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  count and use whole numbers
  
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  use models to represent place value to the tens place
  
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  order and compare numbers (ordinal and cardinal)
  
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  compare whole numbers to landmark numbers (e.g.,10, 25, 50, 75, 100)
  
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  show the relationship between whole numbers (e.g., 1 more/less, 10 more/less)
  
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  compose and decompose (e.g.,14=7+7; 14=10+4)
  
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  apply properties of numbers (odd, even)
  
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  connect number words and numerals to the quantities they represent
  

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  identify common fractions, such as one half, one third, and one fourth
  

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  identify a fraction as part of a whole (i.e., fraction of a set, fraction of an area)
  

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  identify penny, nickel, dime, quarter, and their monetary value
  
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  represent the coin equivalent up to $1.00 using single coins and combinations of coins
  
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  represent monetary values using dollar notation
  

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  count and use whole numbers
  
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  use models and expanded notation to represent place value to the hundreds place
  
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  order and compare numbers (ordinal and cardinal) to each other or to benchmark whole numbers (10, 25, 50, 75, 100, 150)
  
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  compare whole numbers to landmark (benchmark) numbers (e.g., 25, 50, 100, 500)
  
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  show the relationship between whole numbers (e.g., 1 more/less, 10 more/less)
  
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  apply the concepts of equivalency when composing or decomposing numbers (e.g.,34 = 17+17; 34 = 29+5; 34 = 30+4)
  

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  represent common fractions, such as halves, thirds, and fourths
  
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  represent a fraction as part of a whole (i.e., fraction of a set, fraction of an area)
  

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  add coins (to 1.99) to demonstrate understanding of monetary value
  
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  represent monetary value using dollar notation
  
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  make change from $1.00 or less
  
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  show equivalent coin combinations of the same value ( to $1.99)
  

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  find equivalent representations for the same number
  
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  use models, explanations, or other representations to represent place value to the thousands place (1-1; 1-2)
  
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  order and compare numbers (ordinal and cardinal) (1-5; 1-6)
  
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  compare whole numbers to landmark (benchmark) numbers (e.g., 100, 250, 500, 1,000) (1-5)
  
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  show the relationship between whole numbers
  
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  apply the concept of equivalency when composing and decomposing numbers (e.g., 34 = 17+17; 34 = 29+5; 34 = 30+4)
  

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  represent common fractions, such as halves, thirds, fourths, sixths, and eighths (12-1; 12-2; 12-3; 12-4)
  
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  represent a fraction as part of a whole (i.e., fraction of a set, fraction of an area) (12-1)
  
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  compare or identify equivalent positive fractional numbers (e.g., a/2, a/3, a/4) where a is a whole number greater than 0 and less than or equal to the denominator) (L12-5)
  
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  use models, number lines, or explanations (L12-6; 12-7; 12-8; 12-9)
  

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  represent decimals as part of 100 (within the context of money
  

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  find equivalent representations for the same number
  
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  generate whole numbers by decomposing and composing numbers (e.g., 786=7x100 plus 8x10 plus 6x1)
  
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  use models to represent place value through the ten thousands place (1-1; 1-2)
  
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  order or compare whole numbers (1-3)
  

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  recognize common fractions such as half, thirds, fourths, fifths, sixths, eighths, and
  

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  recognize fractions as parts of a collection or of a whole
  
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  recognize locations on number lines
  
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  recognize division of whole numbers
  
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  use linear models where the number of parts in the whole are equal to, and a multiple or factor of the denominator
  
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  use models, benchmarks and equivalent forms to judge the size of fractions (e.g., a/2, a/3, a/4, a/5, a/6, a/8/, a/10)
  
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  recognize and generate equivalent forms of commonly used fractions
  
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  order, compare, or identify equivalent proper positive fractional numbers
  

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  order and compare decimals (12-2)
  
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  relate decimals within the context of money as part of 100 (1-5)
  
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  convert percents as part of 100***
  
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  compute decimals within the context of metric measurement as part of 10^th (e.g., 2.3 cm.) ***
  
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  apply place value structure of decimals (12-1)
  
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  recognize and generate equivalent forms of commonly used decimals and percents ***
  

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  formulate rational numbers from 0 to 9,999,999 through equivalency, composition, decomposition or place value using models, explanations, or other representations [Topic 1]
  
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  recognize the place value structure of the base ten number system (Mathematical Thinking, Inv. 2, 3, 4; or other resource) [Topic 1]
  

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  solve problems involving the use of properties of factors, multiples, prime, or composite numbers [Topic 3]
  
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  determine that the number of parts in the whole is equal to 100, a multiple of 100, or a factor of 100 [Topic 3]
  

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  investigate integers in context using models or number lines [Topic 4,17]
  
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  explore numbers less than 0 by extending the number line [Topic 17]
  

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  recognize the relative magnitude of numbers by ordering, comparing, or identifying equivalent positive fractional numbers within number formats (i.e., fractions to fractions, decimals to decimals, or percents to percents) [Topic 9 ]
  

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  represent fractions as halves, fourths, eighths, thirds, sixths, twelfths, fifths or powers of ten (e.g., 10, 100, 1000) [Topic 9 ]
  
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  use positive fractional numbers (i.e., proper, mixed numbers, and improper) with unlike denominators [Topic 10 ]
  

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  recognize the relative magnitude of numbers by ordering, comparing, or identifying equivalent positive decimals within number formats (i.e., fractions to fractions, decimals to decimals, or percents to percents) [Topic 1,9 ]
  
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  determine that the number of parts in the whole is equal to the denominator of the fractional equivalent of the decimal, or a factor of the denominator of the fractional equivalent of the decimal [Topic 9 ]
  
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  compare whole numbers and decimals [Topic 1 ]
  
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  use decimals to the hundredths and thousandths (including powers of ten) ) in problem solving, or models, or other representations [Topic 9 ]
  

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  recognize the relative magnitude of numbers by ordering, comparing, or identifying equivalent positive benchmark percents (10%, 25%, 50%, 75%, 100%) within number formats (i.e., fractions to fractions, decimals to decimals, or percents to percents) [Topic 16 ]
  
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  recognize benchmark percents (e.g., 10%, 25%, 50%, 75%, or 100%) as parts to a whole using models, explanations, or other representations[Topic 16]
  

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  define ratios and write ratios in three forms (Ch. 7-1)
  
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  compare and contrast ratios and fractions (Ch. 7-1)
  
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  represent part to whole relationships (Ch. 7-2)
  
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  represent decimal numbers on a 10 by 10 grid
  
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  relate fraction benchmarks to decimal benchmarks
  
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  use division to change a fraction to a decimal (Ch. 4-9)
  
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  find equivalent forms of fractions (using common denominators)
  
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  use hundredths grids to develop an understanding of percent as meaning out of 100
  
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  show equivalence among fractions, decimals, and percents (Ch. 7-6)
  
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  express sets of data containing items other than 100 as a percent (Ch. 7-6, 7-7)
  
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  use concrete models to compare fractions and find equivalent fractions
  
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  label points on the number line between whole numbers
  
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  write, compare (using equality & inequality symbols), and order decimals with place values to the ten thousandths (Ch. 1-5, 1-6, 12-2)
  
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  use factors, multiples, prime factorization, and relatively prime numbers to solve problems (Ch. 4-3, 4-4, 4-7)
  
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  use a number line to develop meaning and represent integers (Ch. 11-7)
  
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  calculate exponents and define the base and power numbers (Ch. 4-2)
  

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  demonstrate equivalency among representations of fractions, decimals, and percents (Ch. 2-6, 6-2)
  
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  represent and compare quantities with integers
  
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  compare and order rational numbers (fractions, decimals, whole numbers, whole numbers with exponents, integers, absolute values, numbers written in scientific notation) using the number line or inequality symbols (1-6, 2-4 Teachers must make sure to include all types of numbers in the comparisons as new ideas are learned.)
  
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  calculate and compare unit rates (5-2)
  
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  use proportional reasoning to find the missing value in a proportion (5-4 Teachers must make sure students understand proportionality and have a variety of strategies to show understanding.)
  
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  use percent to represent multiples of a number (6-4)
  
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  find values for percents of solve problems with numbers greater than 100% and less than 1% (6-3)
  
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  use ratios, proportion, and percents to compare two sets of data (5-1, 5-2, 5-3, 5-4, 6-4, 6-5, 6-6, 6-7)
  
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  write large numbers in scientific notation and vice versa (2-8)
  
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  use calculators to work with large numbers and scientific notation (2-8)
  
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  represent and compare quantities with integers (1-6)
  

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  calculate percent increase and percent decrease, relative to the original amount(5-5, 5-6)
  
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  use percent to describe change(5-5, 5-6)
  
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  use absolute value to solve problems(1-2)
  
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  use models to represent perfect square and cube roots(Teachers should use this modeling in 3-1)
  
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  compare and order rational numbers using the number line or inequality symbols(2-3, 2-7, 2-8 Teachers must make sure to include all types of numbers in the comparisons as new ideas are learned.)
  

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  use words such as more than, less than, and add/subtract to express some number concepts
  
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  use numbers to represent real life story problems
  

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  use addition and subtraction of whole numbers in a variety of situations
  
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  describe the relationship between addition and subtraction
  
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  describe the effects of adding and subtracting whole numbers
  

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  describe various meanings of addition and subtraction of whole numbers and the relationship between the two operations
  
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  describe the effects of adding and subtracting whole numbers
  
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  describe situations that entail multiplication and division, such as equal groupings of objects and sharing equally
  

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  use addition or subtraction of positive fractional numbers with like denominators (12-8; 12-9)
  
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  use addition or and subtraction of decimals (13-3)
  
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  describe or illustrate the inverse relationship between addition and subtraction or multiplication and division (3-1; 3-3; 8-1)
  
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  describe the relationship between repeated addition and multiplication using models, number lines, or explanations (5-1)
  
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  accurately solves problems involving addition and subtraction using algorithm as one method with regrouping
  

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  accurately solves problems involving the concept of multiplication and addition or subtraction of decimals in the context of money (T-6; T-7; T-8)
  

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  apply the conventions of order of operations where the left to right computations are modified only by the use of parenthesis
  
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  accurately solve problems involving multiple operations on whole numbers or the use of the properties of factors and multiples (multiplication limited to 2 digits by 2 digits and division to 1 digit divisors) (T-3; T-4; T-5; T-13T-18)
  
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  describe or illustrate the relationship between repeated subtraction and division (no remainders) (L4-1)
  
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  convert the inverse relationship between division and multiplication of whole numbers (e.g., 5 x 2; 2 x 5)(T-3; T-4)
  
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  determine the effects of multiplying and dividing whole numbers (T-3; T-4; T-5; T-7; T-8)
  
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  use properties of operations
  

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  solve problems using addition or subtraction of fractions (proper) [Topic10]
  

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  explain the effects of multiplying and dividing whole numbers [Topic 3,4 ]
  
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  identify and use relationships between operations, such as the divisor as the inverse of multiplication to solve problems [Topic 4]
  
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  use properties of operations (i.e., the distributive property of multiplication and addition)(e.g.,7x23 can be 7x10 plus 7x3) [Topic 1,3,6 ]
  
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  use mathematical operations to describe or illustrate the meaning of a remainder with respect to division to whole numbers using models, explanations or solving problems (i.e., division of whole numbers by up to a 2 digit divisor) [Topic 4,5-4,5-5,5-6 ]
  
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  solve problems involving multiple operations with whole numbers by applying the conventions of order of operations with and without parenthesis [Topic 3,6 ]
  
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  apply properties of numbers (odd, even, divisibility) and field properties(associative, identity, commutative, distributive) to solve problems and computation [Topic 3 ]
  

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  solve problems involving adding and subtracting decimals [Topic 4 ]
  
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  solve problems involving decimals to the hundredths place [Topic 4]
  

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  add, subtract, multiply & divide fractions and decimals (Ch. 5, 6, 1-7, 1-8, 1-9)
  
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  represent fractions larger than a whole using mixed numbers and improper fractions (Ch. 4-6)
  
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  add and subtract integers (Ch. 11-3, 11-4)
  
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  represent fractions with denominators of 10 and powers of 10 as decimal numbers
  
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  use percent in estimating, tax, tip, and discounts (Ch. 7-9)
  
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  describe or illustrate the meaning of power and how it relates to the base and exponent of a number in exponential form (Ch. 4-2)
  
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  describe or illustrate the effect that multiplying or dividing a whole number, decimal, or fraction has on a whole number (Ch. 1, 5, 6)
  
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  solve problems involving single or multiple operations on fractions (proper, improper, and mixed) or decimals by correctly applying the Order of Operations with and without parentheses (Ch. 4-6)
  
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  solve problems involving single or multiple operations by adding or subtracting integers or percent of a whole (Ch. 7-7, 11-7)
  
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  solve problems involving greatest common factor or least common multiple (Ch. 4-4, 4-7)
  
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  classify whole numbers greater than one as prime or composite (Ch. 4-3)
  
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  classify numbers as odd or even (Ch. 4-1)
  
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  apply divisibility rules for 2, 3, 5, 9, and 10 (Ch. 4-1)
  
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  determine the prime factorization of a number using a factor tree (Ch. 4-3)
  
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  apply commutative, associative, distributive, additive identity, multiplicative property of one and additive inverse properties Ch. 1-3, 3-4, 3-8)
  

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  add, subtract, multiply and divide integers (1-7 Teachers must make sure students have a strong understanding of how integer values are combined and the difference between them.)
  
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  solve problems using proportional reasoning (5-2, 5-3, 5-4, 5-5, 5-6)
  
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  solve problems involving discounts, tax, tip, rates (6-7)
  
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  apply order of operations including parentheses, brackets, and exponents (1-9, 2-1 Teachers must make sure to provide problems using a combination of all forms.)
  
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  find values of numbers with whole number exponents (6-8)
  
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  determine the square root of a number (8-6)
  
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  use the inverse relationships of addition and subtraction; multiplication and division; and squaring and finding square roots to simplify computations and solve problems (4-3, 4-4, 8-6 Teachers must incorporate this thinking into problem solving in all areas of mathematics.)
  
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  apply commutative, associative, distributive, identity and inverse properties (1-2, 1-3, 1-9 Teachers must incorporate these properties when solving problems in all areas of mathematics.)
  

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  use practical problems to review and strengthen understanding of the order of operations (1-1)
  
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  add, subtract, multiply and divide rational numbers (2-4, 2-5)
  
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  use models and diagrams to represent operations with integers and whole number exponents
  
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  generalize and use the distributive and commutative properties to solve problems (1-5)
  
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  solve problems requiring inverse operations (Chapter 6 Teachers must imbed this thinking throughout all areas of mathematics problem solving.)
  
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  relate the area of a square to the length of its side (Teachers should use this modeling with students in 3-1)
  
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  use the associative and commutative properties of addition and multiplication and the distributive property to simplify computations with integers, fractions, and decimals (2-4, 2-5)
  
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  use multiplication and division of integers, square roots, cube roots, squaring and cubing to solve problems involving percent increase and decrease, interest rates, and proportional relationships (4-3, 4-4, 5-3, 5-4, 5-5, 5-6, 5-8, 5-9)
  
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  use the additive inverse (any number added to its opposite is equal to zero), multiplicative inverse (any number multiplied by its reciprocal is equal to one), identity element for multiplication (any number multiplied by one remains unchanged), identity element for addition (any number added to zero remains unchanged) to solve problems (chapter 6)
  
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  recognize that any number raised to the power of zero is equal to one (2-7)
  

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  reasonably estimate quantities less than 20
  

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  develop and use strategies for whole-number computations, with a focus on addition and subtraction
  
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  develop fluency with basic number combinations for addition and subtraction to 20 (same units)
  
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  use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil
  

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  develop and use strategies for whole-number computations, with a focus on addition and subtraction
  
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  develop mastery of basic number combinations for addition and subtraction to 20
  
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  use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators.
  
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  Reasonably estimate quantities less than 50
  

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  develop fluency in adding, subtracting, multiplying and dividing whole numbers T2-3; 2-6; T3-3; T4-2,3,4; T5-6,7,8,9; T8-2,3,4,5
  
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  develop and use strategies to estimate the results of whole number computations T 2-5; T3-4
  
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  select appropriate methods and tools for computing
  
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  develop and use strategies to estimate computations involving fractions in situations relevant to students’ experience ***
  
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  use models, benchmarks, and equivalent forms to add and subtract commonly used fractions T1 2-8; T-9
  
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  develop and use strategies to estimate computations involving decimals in situations relevant to students’ experience
  
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  use models, benchmarks, and equivalent forms to add and subtract commonly used fractions
  

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  use basic number combinations for multiplication and division to mentally compute related problems; multiplies whole number facts to a product of 100 and calculates related division facts
  
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  develop fluency in adding, subtracting, multiplying, and dividing
  

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  use strategies to estimate the results of whole number computations to judge the reasonableness of such results
  
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  use appropriate tools and methods for computing with whole numbers; mentally adds two-digit whole numbers, combinations of two-digit and three digit whole numbers that are multiples of ten, and four-digit whole numbers that are multiples of 100 (e.g., 67+24, 320+430, 1300+1400)
  
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  use models, benchmarks, and equivalent forms to add and subtract positive, proper, commonly used fractions with like denominators
  
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  use strategies to estimate computations involving fractions in situations relevant to students’ experiences
  
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  mentally add and subtract whole number facts through 20, subtract a one-digit whole number from a two-digit whole number (e.g., 67-9) and combinations of two-digit and three-digit whole numbers that are multiples of ten (e.g., 50-30, 230-80, 520-200)
  

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  mentally compute basic number combinations for multiplication (e.g. 30x50, 45x5, 400 x50, 400x600, 360÷12, 360÷6, 360÷60, 3600÷6) and division [Topic 3,4 ]
  
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  mentally compute problems to a product of 144[Topic 3 ]
  
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  fluently add, subtract, multiply, and divide whole numbers [Topic 2,3 ]
  
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  use strategies to estimate the results of whole number computations and to judge the reasonableness of such results [Topic 2 ]
  
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  use strategies to estimate computations involving fractions and decimals relevant to students’ experience [Topic 7,9 ]
  
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  mentally calculate change back from $1.00, $5.00, and $10.00 [Topic 2,4 ]
  
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  mentally multiply 2-digit whole numbers by a one-digit whole number (e.g. 45x5) [Topic 3 ]
  
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  mentally calculate two digit whole numbers that are multiples of ten (50 x 60) [Topic 3 ]
  
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  mentally calculate a three digit whole number that is a multiple of 100 by a two or three digit number which is a multiple of 10 or 100 (ex. 400x50, 400x500) [Topic 3]
  
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  mentally divide 3 and 4 digit multiples of powers of ten by their compatible factors (ex. 360÷6, 360÷60,3600÷600) [Topic 4]
  

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  develop strategies for estimating sums of fractions and decimals
  
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  use benchmark fractions to estimate sums (e.g., ¼, ½, 1 ½)
  
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  generalize patterns to develop algorithms for adding, subtracting, multiplying, and dividing fractions
  
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  generalize patterns to develop algorithms for adding, subtracting, multiplying, and dividing decimals
  
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  generalize patterns to develop algorithms for adding and subtracting integers
  
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  develop and use strategies to estimate the results of a solution
  
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  select when mental computation, estimation, calculators, or pencil and paper strategies would be most appropriate
  
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  use percent in estimating tax, tip, and discounts
  
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  mentally calculate change back from $20.00, $50.00, and $100.00
  
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  mentally multiply a two-digit number by a one-digit number
  
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  mentally multiply two-digit numbers that are multiples of ten
  
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  mentally multiply a three-digit number that is a multiple of 100 by a two- or three-digit number that is a multiple of 10 or 100
  
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  mentally divide three- and four-digit multiples of powers of ten by their factors
  
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  mentally determine parts of a whole number using the benchmarks 1%, 10%, 25%, 50%, and 75%
  

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  develop and use strategies to estimate the results of a solution
  
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  use mental calculations and benchmarks of fractions, percentages, perfect squares, and square roots
  
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  use estimation to determine the reasonableness of an answer
  
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  compute fluently with fractions, decimals, and integers (Chapters 1, 2, 3)
  
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  select when mental computation, estimation, calculators, or pencil and paper strategies would be most appropriate
  
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  develop and use strategies to estimate the results of a solution
  
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  recognize when ratios are a useful form of comparison
  
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  apply proportional reasoning to real world situations
  

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  decide when to use mental computation, estimation, calculators, or pencil and paper depending on the situation
  
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  develop and use strategies to estimate the results of a solution
  
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  select and apply appropriate strategies to make comparisons
  
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  use mental calculations and benchmarks of fractions, percentages, perfect squares, and square roots
  
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  use estimation to determine the reasonableness of an answer
  
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  compute fluently with fractions, decimals, and integers (Chapter 2)
  
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  recognize when ratios are a useful form of comparison
  
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  apply proportional reasoning to real world situations
  

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  sort, classify and order objects by one attribute (e.g., size, number, or other property)
  
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  recognize, describe, and extend a simple repeating pattern (e.g., ABAB) or growing pattern (e.g., 1, 2, 3) of shapes, sounds or numbers.
  

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  sort, classify, and order objects by one or two attributes (e.g., size, number, and other properties)
  
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  recognize, describe and extend a repeating pattern (e.g., ABAB) or growing pattern (e.g., 10, 20, 30) of shapes, sounds or numbers
  
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  recognize the same pattern in different situations (e.g., AB = slap, clap = red, blue)
  

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  sort, classify, and order objects and numbers by one or two attributes (i.e., size, number, odd/even numbers, multiples of 5 or 10)
  
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  identify and extend a variety of patterns ( i.e., linear and non-numeric) represented in models, tables, or sequences (e.g., 2, 4, 6, __, 10)
  
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  recognize and compare similar repeating or growing patterns in different representations (e.g., AB, slap-clap, red, blue)
  

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  identify specific cases of a variety of patterns both linear and non-numeric, represented in models, tables, words, graphs or sequences by extending the pattern to the next one, two, or three elements, or finding missing elements (geometric and numeric patterns)
  

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  identify and extend to specific cases, a variety of patterns (i.e., linear and nonlinear) represented in models, tables, words, graphs, or sequences; writes a rule in words or symbols to find the next
  

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  identify and extend to specific cases a variety of patterns (i.e., linear and nonlinear) represented in models, tables, words, graphs, sequences or in problem situation [Topic 6 ]
  
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  write a rule in words or symbols for finding specific patterns in cases of a linear relationship [Topic 6 ]
  

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  analyze data using coordinate graphs to explore relationships among variables (Ch. 11-10)
  
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  explain the difference between a numerical expression and an algebraic equation (Ch. 3-2)
  
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  identify a variety of linear or nonlinear patterns represented in models, tables, sequences, graphs, and problem situations (Ch. 3-1, 3-3)
  
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  write an expression or equation using words or symbols to express a generalized linear relationship (Ch. 3-1, 3-2, 3-3)
  

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  interpret information given in a table or graph
  
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  search for patterns of predictable change
  
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  extend a pattern when represented in the form of a table, graph, sequence, model, or problem situation
  
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  write a general, explicit rule for a linear or nonlinear sequence when represented in table, graph, model, or problem situation
  
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  find a specific case of a linear relationship
  
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  recognize a function as linear or nonlinear when represented in table, graph or equation form
  
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  use a graphing calculator to generate a table or graph given a rule or situation
  

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  recognize linear situations in written descriptions, tables, graphs, and symbols (11-3, 11-4, 11-5, 11-6)
  

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  recognize how the y-intercept appears in tables, graphs, and equations (11-3, 11-4, 11-5, 11-6)
  
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  recognize how the rate of change (slope) appears in tables and equations and affects the graph of a line (11-3, 11-4, 11-5, 11-6)
  
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  use data from tables, graphs and equations interchangeably (11-3, 11-4, 11-5, 11-6)
  
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  identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations (11-7)
  
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  distinguish between linear and nonlinear relationships
  
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  write a formula, using symbols, for a linear sequence presented in model, table, graph or problem solving situation (11-1)
  
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  generalize a rule using a formula or words for a nonlinear sequence presented in model, table, graph or problem solving situation (Chapter 11)
  

N/A

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  investigate general properties of addition and subtraction (i.e., commutative, zero property) using arithmetic notation along with objects and pictures
  

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  illustrate general principles and properties of addition and subtraction (i.e., commutative property, zero properties, associative properties) using specific numbers
  
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  show a conceptual understanding of equality in addition and subtraction expressions by finding a value that makes an open sentence true (e.g.3+ _ = 6) (limited to one operation and limited to use addition or subtraction)
  
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  use concrete, pictorial, and verbal representation for invented and conventional arithmetic notations
  

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  show the equality between two expressions using models or different representations of the expressions or by finding the value that will make an open sentence true (e.g., 2 + _ = 7) using one operation with addition, subtraction or multiplication
  
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  solve mathematical problems to show relationships with equations
  

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  use algebraic expressions by using letters or symbols to represent unknown quantities to write simple linear algebraic expressions involving any one of the four operations
  
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  evaluate simple linear algebraic expressions using whole numbers
  
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  identify commutative, associative, and distributive properties to compute with whole numbers
  
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  solve mathematical relationships using equations
  

• 
  develop algebraic expressions by using letters or a symbol to represent unknown quantities to write linear algebraic expressions involving any two of the four operations; or by evaluating linear algebraic expressions using whole numbers [Topic 6 ]
  
• 
  identify commutative, associative, and distributive properties and use them to compute with whole numbers [Topic 6 ]
  

• 
  substitute a variable in situations where a rule can be applied such as C=2r ( Ch. 9)
  
• 
  recognize and generate equivalent forms for a simple algebraic expression and solve linear equations (e.g., P=2l + 2w or P=2(l+w)) ( Ch. 9)
  
• 
  use letters to represent unknown quantities to write simple algebraic expressions using four basic operations and following the Order of Operations ( Ch. 3-3)
  
• 
  evaluate algebraic expressions with more than one variable ( Ch. 3-2)
  
• 
  evaluate an algebraic expression within an equation ( Ch. 3-5,3-6,3-7)
  
• 
  accurately constructs/interprets coordinate graphs ( Ch. 11-8)
  
• 
  plot points, identify origin and four quadrants and read coordinates from graph ( Ch. 11-8)
  
• 
  describe the meaning of slope and intercept in concrete situations ( Ch. 11-10)
  
• 
  describe how change in the value of one variable relates to the change in the value of a second variable in problem situations with constant rates of change ( Ch. 11-10)
  

• 
  examine the relationships between tables, graphs, and equations (10-2)
  
• 
  describe the meaning of slope given a concrete situation (10-3)
  
• 
  solve problems using slope and rate of change (10-3)
  
• 
  determine slope from a table or graph (9-2)
  
• 
  describe how the change in one variable relates to the change in a second variable (in linear relationships)
  
• 
  use letters to represent unknown quantities when writing algebraic expressions and equations (including using whole number exponents or more than one variable) (4-1)
  
• 
  solve linear equations in the form of y= mx + b (4-5, 4-6)
  
• 
  evaluate expressions or equations through substitution (4-1)
  

• 
  recognize that a change in rate will change the steepness of a line and the coefficient of x (11-4)
  
• 
  represent linear relationships in tables, graphs, formulas
  
• 
  translate among representations of linear relationships
  
• 
  find the slope and intercepts of the graph of a linear relationship and identify their meaning in a problem situation
  
• 
  describe how one quantity varies in relationship to another quantity when there is a constant (linear) or varying (nonlinear) rate of change
  
• 
  solve an equation of the form y=mx + b (6-1, 6-3)
  
• 
  simplify algebraic expressions (6-2)
  
• 
  evaluate expressions within an equation (6-2, 6-3)
  
• 
  solve linear equations through substitution (1-1, 1-4, as a check when solving equations in chapter 6)
  
• 
  recognize and generate equivalent expressions (example: re-express formulas such as d=rt in terms of t as t=d/r) (2-6)
  
• 
  show equivalence or nonequivalence of 2 or more expressions by applying commutative, associative, distributive properties, order of operations, or substitution (6-2, 6-3, 6-4)
  
• 
  show equivalence between two expressions using models or other representations
  
• 
  solve multi-step linear equations with integer coefficients (6-3, 6-4)
  
• 
  solve problems informally involving systems of linear equations in context
  

• 
  use objects and pictures to model real life situations involving joining actions (i.e., addition) and separating actions (i.e., subtraction)
  

• 
  use objects, pictures and symbols to model situations involving joining actions and separating actions of two to four numbers using one operation
  
• 
  write an equation that models a problem situation
  
• 
  Find the value that will make an open sentence true)e.g. 2+__=7) using models, verbal explanations, or written equations
  

• 
  use objects, pictures, and symbols to model situations involving joining actions and separating actions of whole numbers; both operations may be used in a single equation
  
• 
  use more than one model to demonstrate an addition or subtraction problem
  
• 
  use tables or equations to model a problem situation
  
• 
  find the value that will make an open sentence true)e.g. 2=__7) using models, verbal explanations, or written equations
  

• 
  model problem situations with objects and use graphs, tables, equations to draw
  
• 
  find the value that will make an open sentence true)e.g. 2=__7) using models, verbal explanations, or written equations
  

• 
  show the concept of equality between two expressions using models or different representations of the expressions such as with objects, graphs, tables, and equations to draw conclusions
  
• 
  show equivalence by simplifying numerical expressions where left to right computations may be modified only by the use of parenthesis [e.g., 14-(2x5)] (expressions consistent with simple linear algebraic expressions involving any of the four operations or using whole numbers) (***)
  
• 
  solve one-step linear equations of the form ax = c x≠ to b = c, where a, b, and c are whole numbers with a ≠ to o
  

• 
  show equivalence between two expressions using models or different representations of the expressions (using letters to represent unknown quantities to write linear algebraic expressions involving any two of the four operations or using whole numbers)by solving one step linear equations (of the forms ax=c, x≠b=c, or x/a=c, where a, b, and c are whole numbers with a≠0)[Topic 6 ]
  
• 
  show equivalence by determining which values of a replacement set make the equation (multi-step of the form ax + b = c where a, b, and c are whole numbers with a 0) a true statement (e.g., 2x+3 = 11, x:x=2, 3, 4, 5})
  

• 
  model problems with objects, graphs, tables, or equations
  

• 
  model problems with graphs, tables, or equations
  
• 
  show equivalence between two expressions using models or different representations (e.g., using letters to represent unknown quantities to write linear algebraic expressions involving any of the four operations and consistent with order of operations) ( Ch. 12-1)
  
• 
  solve multi-step linear equations using inverse operations ( Ch. 3-5, 3-6, 3-7)
  

• 
  translate a problem solving situation into an equation (4-1, 4-3, 4-4, 4-5, 4-6)
  
• 
  show equivalence between two expressions (using models or other representations) (4-3)
  
• 
  collect data from an experiment, then represent the data using a table and graph (Chapter 9)
  
• 
  create a table, make a graph, and determine a rule for a given situation (Chapter 9)
  
• 
  collect data and use patterns in tables and graphs to make predictions (Chapter 9)
  

• 
  model problems (linear and nonlinear) using tables, graphs, equations, and words (Chapter 11)
  
• 
  show equivalence between two expressions using models or other representations (Chapter 11)
  
• 
  create a table, make a graph, and determine a rule, based on a constant or varying rate of change, for a given situation (Chapter 11)
  
• 
  solve contextualized problems (linear and nonlinear) represented with graphs, tables, equations and words (Chapter 11)
  

• 
  describe how things change over time using real world models (e.g., we grow taller, the weather gets colder)
  

• 
  associate numbers with change (e.g., The plant is growing two more inches each week)
  

• 
  describe and predict both qualitative and quantitative changes in the real world
  

• 
  Grade Three M(DSP) 3-3;3-6 (T20)
  
• 
  introduce how a change in one variable relates to a change in a second variable
  
• 
  identify, describe, and/or compare situations that represent constant rates of change
  

• 
  investigate how a change in one variable relates to a change in a second variable
  
• 
  identify and describe situations with constant or varying rates of change and compare them
  

• 
  Compare and describe situations that represent constant or varying rates of change (e.g. tell a story given a line graph about a trip) in a linear relationship (y=kx) as a constant rate of change [Topic 18]
  

• 
  write a rule in words or symbols for linear or nonlinear relationships ( Ch. 3-1)
  
• 
  construct and interpret graphs of real-life occurrences ( Ch. 11-10)
  
• 
  describe the slope of a line for linear relationships (e.g., faster, slower, greater, or smaller) ( Ch. 11-10)
  

• 
  use graphs to analyze the nature of changes in quantities of linear relationships (9-5)
  
• 
  determine the slope of a line from a table or graph (10-3)
  

• 
  recognize rate of change in linear relationships from tables, graphs, and equations (Chapter 11)
  
• 
  find slope and intercepts of the graph of a linear relationship and identify their meaning in a problem situation (Chapter 11)
  
• 
  describe how one quantity varies in relationship to another quantity when there is a constant (linear) or varying (nonlinear) rate of change (Chapter 11)
  
• 
  relate the slope and y-intercept to the equation of the line (Chapter 11)
  
• 
  use knowledge about linear relationships to solve problems (Chapter 11)
  

• 
  name basic two-dimensional and three-dimensional shapes and describe some of their attributes (e.g., square, triangle, circle, rectangle, side, top, corner)
  
• 
  make simple comparisons between shapes (e.g., bigger, taller, longer, wider, etc.) and group shapes that go together
  

1. 
   create a design using manipulatives (e.g., pattern blocks)
   
2. 
   classify, sort, and compare shapes using one attribute
   
3. 
   build polygons and circles
   

• 
  name and describe attributes and parts of two-dimensional and three-dimensional shapes (e.g., trapezoid, hexagon, rhombus, size, number of sides and corners)
  
• 
  compare shapes by at least 1 attribute (e.g., size, number of sides, number of corners, etc.)
  
• 
  compose (combine) and decompose shapes using manipulatives (e.g., Pattern block puzzles)
  
• 
  distinguish between simple geometric shapes and non-examples (i.e., Which are triangles and which are not triangles?) ***
  

• 
  identify and describe two-dimensional and three-dimensional shapes using mathematical terms (e.g., sides, faces, angles, etc.)
  
• 
  sort and classify two-dimensional and three-dimensional shapes by two or more attributes (e.g., sides, shapes, angles, etc.)
  
• 
  predict results of putting together (composing) and taking apart (decomposing) shapes
  
• 
  distinguish between geometric shapes and non-examples (i.e., Which are rectangles and which are not rectangles?)
  

• 
  identify, compare, and analyze attributes or properties of angles (i.e., number of angles) or sides (i.e., number of sides or length of sides) of two and three-dimensional shapes
  
• 
  develop vocabulary to describe attributes and define classes of shapes such as triangles and pyramids
  
• 
  use composition or decomposition of shapes to identify, describe, or distinguish among triangles, squares, rectangles, rhombi, trapezoids, hexagons, or circles
  
• 
  demonstrate congruence and similarity of shapes
  

• 
  identify, compare, and analyze attributes of angles (i.e., number of angles) or sides (i.e., number of sides, length of sides, parallelism or perpendicularity)
  
• 
  identify, describe or distinguish among triangles, squares, rectangles, rhombi, trapezoids, hexagons, or octagons
  
• 
  classify angles relative to 90 degrees as more than, less than, or equal to
  
• 
  identify, compare, or describe three-dimensional shapes (e.g., rectangular prisms, triangular prisms, cylinders, or spheres)
  
• 
  demonstrate congruence of geometric shapes using reflections, translations, or rotations such as flips, slides, turns
  
• 
  make and test conjectures about geometric properties
  
• 
  describe similarity (i.e., apply scales on maps, apply characteristics of same shape but different proportional size, to identify similar figures)
  
• 
  compose or decompose shapes using models or explanations
  

• 
  identify, compare, and analyze attributes of angles (right, acute, or obtuse) or sides (number of congruent sides, parallelism, or perpendicularity) to identify, describe, classify, or distinguish among different types of triangles (right, acute, obtuse, equiangular, or equilateral) or quadrilaterals (rectangles, squares, rhombi, trapezoids, or parallelograms) [Topic 8]
  
• 
  use properties (numbers of bases, shape of bases, number of lateral faces) to identify, compare, describe, or build models of three-dimensional shapes (rectangular prisms, triangular prisms, cylinders, spheres, pyramids, or cones)[Topic 13]
  
• 
  describe the proportional effect on the linear dimensions of triangles and rectangles when scaling up or down while preserving angle measures, or by solving related problems (including applying scales on maps) using models or explanations. [Topic 8/Social Studies Curriculum]
  

• 
  use properties of sides and angles in isosceles and equilateral triangles and parallelograms to solve problems
  
• 
  identify, describe, or classify triangles by lengths of sides (equilateral, isosceles, or scalene) or by measures of angles (right, acute, obtuse, or equiangular)
  
• 
  identify, describe, or classify quadrilaterals (rectangles, squares, rhombi, trapezoids, or parallelograms) by lengths of sides, measures of angles, number of congruent sides, parallelism, or perpendicularity
  
• 
  identify, compare, or describe three-dimensional shapes (rectangular prisms, triangular prisms, cylinders, spheres, pyramids, or cones) using three-dimensional models and two-dimensional representations
  
• 
  identify bases, shape of bases, number of lateral faces, number of bases, number of edges, and number of vertices in the above-mentioned three-dimensional shapes
  
• 
  recognize the relationships among similar figures by drawing similar and non-similar figures (e.g., angles, side lengths, perimeter)
  
• 
  recognize corresponding angles
  
• 
  identify, describe, or classify angles by their measure (right, acute, or obtuse)
  
• 
  draw similar and non-similar figures on a coordinate grid
  

• 
  show that any two similar figures are related by a scale factor (5-5, 5-6)
  
• 
  recognize the areas of similar figures grow by the square of the scale factor (While studying scaling, the teacher must be sure to pose problems that help students make this connection.)
  
• 
  use the concept of similar figures to solve problems in the real world (e.g., shadow method, mirror method, congruent triangles) (5-5)
  
• 
  recognize similar figures and be able to tell why they are similar (5-5)
  
• 
  apply properties of similar figures by describing the impact scaling up or down has on angle measures, linear dimensions and areas of polygons and circles (5-5)
  
• 
  use properties of angle relationships (adjacent angles, vertical angles, straight angles) when two or three lines intersect to solve problems (7-2)
  
• 
  use properties of two parallel lines cut by a transversal to solve problems
  
• 
  apply the triangle inequality theorem (sum of the lengths of any two sides of a triangle is greater than the length of the third side) (7-3)
  
• 
  calculate the sum of the interior angles of a regular polygon (for a polygon with n sides, (n-2)180 is the sum of the interior angles of a polygon (page xxxv)
  

• 
  identify and apply the Pythagorean Theorem to find missing lengths on right triangles (3-2, 3-3)
  
• 
  solve problems using properties of angle relationships (adjacent, straight, vertical angles) when two or three lines intersect or two parallel lines are cut by a transversal (7-1, 7-2)
  
• 
  determine the impact on surface area and volume when scaling an object (8-4, 8-9 The teacher must carefully assign problems to help students make this connection.)
  
• 
  recognize that volumes of similar objects grow by the cube of the scale factor (8-9 The teacher must carefully assign problems to help students make this connection.)
  

• 
  describe the arrangement, order, and position of objects (e.g., on top of, underneath, beside, between, etc.)
  
• 
  follow simple navigation directions (e.g., forward, backward, near, next to)
  
• 
  find and describe the location of objects in a picture (i.e., scene)
  

• 
  name and interpret relative positions in two-dimensional and three-dimensional space (e.g., in front, behind, above, etc.)
  
• 
  follow and give navigational directions or right, left, forward, and backward to get a person to a location
  
• 
  find and describe location of objects on a simple map (e.g., neighborhood)
  

• 
  use appropriate language to describe relative positions (e.g., right, left, diagonal, etc.)
  
• 
  navigate in space by giving directions from one place to another (e.g., number of steps, turns)
  
• 
  find and compare(e.g., near, far) locations on a simple coordinate system or map (e.g., north, south, east, west)
  

• 
  Grade Three M(G&M) 3-9; (T20)
  
• 
  apply coordinate systems to specify locations and to describe paths
  
• 
  find location on a coordinate grid using positional words or compass directions
  
• 
  interpret and give directions from one location to another using positional words
  

• 
  make and use coordinate systems to specify locations and to describe paths
  
• 
  find the distance between points along horizontal and vertical lines of a coordinate system
  
• 
  give directions between locations on a map or coordinate grid (first quadrant)
  
• 
  plot points in the first quadrant in context (e.g., games, maps)
  
• 
  find the horizontal and vertical distances between points on a coordinate grid in the first quadrant
  

• 
  make and use coordinate systems to specify locations [Topic 17]
  
• 
  find the distance between points along horizontal and vertical lines of a coordinate system [Topic 17]
  
• 
  interpret and give directions between locations on a map or coordinate grid (all four quadrants) [Topic 17]
  
• 
  plot points in four quadrants in context (e.g., games, mapping, identifying the vertices of polygons as they are reflected, rotated, and translated) [Topic 17/Social Studies/Science Curriculum]
  
• 
  determine horizontal and vertical distances between points on a coordinate grid in the first quadrant [Topic 17/Social Studies Curriculum]
  

N/A

• 
  compare slopes of parallel lines when examining figures (such as parallelograms) on a coordinate plane (10-1 The teacher must carefully assign problems to help students see this relationship.)
  
• 
  use coordinate geometry to examine shapes with pairs of parallel or perpendicular sides (10-1 The teacher must carefully assign problems to help students see this relationship.)
  

• 
  use relationships between slopes of parallel lines and perpendicular lines to solve problems (11-4)
  
• 
  compare slopes of parallel and perpendicular lines when examining figures (such as parallelograms) on a coordinate plane (11-4)
  

N/A

• 
  recognize and match basic two-dimensional shapes in transformations (e.g., slides, flips, and turns)
  
• 
  recognize line symmetry in simple two-dimensional shapes
  

• 
  recognize and create shapes that are transformations (e.g., slides, flips, and turns)
  
• 
  recognize and create shapes that have line (i.e., mirror) symmetry using manipulatives (e.g., pattern blocks, tiles)
  
• 
  use line symmetry to demonstrate congruent parts within a shape
  

• 
  Grade Three M(G&M) 3-4 (T19)
  
• 
  predict and describe the results of sliding, flipping, and turning two-dimensional shapes
  
• 
  determine if two-plane figures are congruent by matching
  
• 
  identify line (mirror) symmetry in two-dimensional models
  

• 
  describe a motion or series of motions that will show that two shapes are congruent
  
• 
  identify line and rotational symmetry in two-dimensional models
  

• 
  match congruent figures using reflections, translations, or rotations (e.g., flips, slides, or turns) or by composing or decomposing shapes [Topic 19]
  
• 
  identify line and rotational symmetry in two- and three-dimensional models [Topic 19]
  

• 
  recognize and describe flips and turns with respect to triangles and parallelograms
  
• 
  describe and produce a transformation
  
• 
  predict and describe reflections, translations, and rotations to show congruence (including degree of rotation)
  
• 
  predict and describe the transformational steps that result from composing and decomposing two- and three-dimensional shapes using models or explanations
  
• 
  recognize and apply line and rotational symmetry to demonstrate congruent parts within a shape
  
• 
  use line symmetry in three-dimensional models to solve problems
  

• 
  describe and produce a transformation (10-5, 10-6, 10-7)
  
• 
  identify corresponding parts of similar figures once the figures have been transformed (slide, flip,, turn, scaling) (10-5, 10-6, 10-7)
  
• 
  apply concepts of congruency using the coordinate plane when solving problems; recognize that any shape translated or rotated placed exactly on top of another are congruent figures (10-5, 10-6, 10-7)
  
• 
  use transformations to examine congruence and similarity (10-5, 10-6, 10-7)
  

N/A

• 
  recognize a shape or pattern of dots using spatial (visual) memory
  
• 
  describe different sides of a three-dimensional object
  
• 
  recognize basic two-dimensional shapes in the natural environment
  

• 
  reproduce images of basic designs of dots and shapes using spatial (visual) memory
  
• 
  recognize shapes from different perspectives (e.g., match front, back, and sides to an object)
  
• 
  recognize geometric shapes (two-dimensional and three-dimensional) in the natural environment
  

• 
  reproduce images of geometric designs based on spatial (visual) memory
  
• 
  recognize and represent shapes from different perspectives (i.e., draw faces of a 3D wooden block)
  
• 
  recognize geometric shapes and structures in the environment and specify their location
  

• 
  build, compare, copy, and draw geometric objects such as triangles, squares, rectangles, rhombi, trapezoids, hexagons, and circles
  
• 
  describe and create mental images of objects, patterns, and paths
  
• 
  identify and build models of rectangular prisms from three-dimensional representations
  
• 
  use geometric models to solve problems in other areas or disciplines
  
• 
  identify, build, and draw a two-dimensional representation of a three-dimensional object
  

• 
  copy, compare, build, and draw models of triangles, squares, rhombi, trapezoids, hexagons, octagons, and circles
  
• 
  create and describe mental images of objects, patterns and paths
  
• 
  identify and build a three-dimensional object from a two-dimensional representation of that
  
• 
  identify and draw a two-dimensional representation of a three-dimensional object
  
• 
  use geometric models to solve problems
  

• 
  build and draw various geometric objects[Topic 8]
  
• 
  build models of rectangle and triangle prisms, cones, cylinders, and pyramids from two-dimensional or three-dimensional representations [Topic 13]
  
• 
  identify and draw a two-dimensional representation of a three-dimensional object [Topic 13]
  
• 
  use geometric models to solve problems in other areas of mathematics and other disciplines as well as to solve problems in everyday life [Topic 13]
  

• 
  sketch angles using benchmarks
  
• 
  find precise angle measures using a protractor
  
• 
  use two-dimensional objects to build a three-dimensional representation (visa versa)
  
• 
  use angles and angle measures in real life applications
  

• 
  sketch three dimensional solids (8-8)
  
• 
  draw nets (flat patterns) of rectangular prisms, triangular prisms, cylinders, pyramids and use these nets as a tool for finding surface area (8-9)
  
• 
  identify the concept of surface area as wrapping an object (8-9)
  
• 
  solve problems to determine volume when given a two-dimensional drawing of a rectangular prism, triangular prism, cylinder, or pyramid; express measurements using appropriate units (8-10)
  
• 
  find the area of a circle when given one of the following measurements: diameter, radius, or circumference (8-5)
  
• 
  find the area and perimeter of composite figures (8-1, 8-3, 8-4)
  
• 
  find the surface area of rectangular prisms, triangular prisms, cylinders, and pyramids; express measurements using appropriate units (8-9)
  
• 
  find the volume of rectangular prisms, triangular prisms, and cylinders; express measurements using appropriate units (8-10)
  
• 
  recognize and apply geometric ideas in relationships in art, science, and everyday life Teachers will incorporate these types of problems.
  

• 
  represent the Pythagorean Theorem using models and algebraic symbols (3-3 and Pythagorean Puzzle lesson)
  
• 
  determine the length of a missing side of a right triangle, through modeling (3-3 and Pythagorean Puzzle lesson)
  
• 
  find unknown heights through similarity (4-7)
  
• 
  solve problems to determine surface area and volume when given a two-dimensional drawing of a rectangular or triangular prism, cylinder, pyramid, or cone; express measurements with appropriate units (8-4, 8-5, 8-6, 8-7)
  

• 
  begin to order, compare, or describe objects according to size, length, height, and weight , temperature, and capacity (more/less)
  
• 
  describe that some events take place in the past, present or future
  

• 
  determine elapsed time relating to calendar patterns (days of the week, yesterday, today, tomorrow)
  
• 
  sequence the events in a day
  
• 
  identify a clock and a calendar as measurement tools
  

• 
  use standard and nonstandard systems of measurement for length, weight, area, volume, and time and temperature
  
• 
  compare and order objects according to these attributes (i.e., longer than, heavier/lighter than)
  

• 
  select the appropriate unit and tool for the attribute being measured
  
• 
  use the unit of one minute and one hour
  
• 
  use calendar time – days of the week, months of the year
  
• 
  use the unit of one inch and one foot ***
  

• 
  recognize measurement attributes: length (to the whole inch and foot and whole centimeter and meter), volume, weight, perimeter, area, and time (to the hour by 15 min. intervals)
  

• 
  compare and order objects according to these attributes; length, volume, weight, perimeter area and time (using numerical values)
  
• 
  understand how to measure using nonstandard and standard units (e.g., straight line, end to end)
  
• 
  select an appropriate unit and tool for the attribute being measured. ***
  

• 
  use equivalencies of 12 inches equals one foot; 100 centimeters equals one meter, and 60 minutes equals one hour when solving problems. ***
  

• 
  compute perimeter of polygons
  
• 
  draw the area of rectangles on grids using a variety of models
  
• 
  express all measures using appropriate units
  
• 
  measure length, weight, and volume and select an appropriate type of unit for measuring
  
• 
  measure with standard units in the customary system
  
• 
  carry out simple unit conversions when solving problems across content strands
  
• 
  use units of measures appropriately and consistently
  

• 
  recognize perimeter of polygons on grids
  
• 
  find the area of rectangles, polygons, or irregular shapes on grids using a variety of models, manipulatives or formulas
  
• 
  apply all measures using appropriate units (benchmarks: hour to 5 minute interval; day; year; 24 hours in 1 day; 7 days in 1 week; 365 days in 1 year; 60 seconds in 1 minute; 60 minutes in 1 hour; C degrees and F degrees to 1 degree; quart to whole quart; kilogram to whole kilogram; gram to whole gram; pound to whole pound)
  
• 
  measure quantities such as length, area, weight, volume and size of angle
  
• 
  measure with standard units in the customary and metric systems
  
• 
  recognize that measurements are approximations and how differences in units affect precision
  
• 
  apply what happens to measurements of a 2-dimensional shape such as its perimeter and area when the shape is changed
  

• 
  calculate the perimeter of polygons using models, manipulatives, or formulas [Topic 12]
  
• 
  calculate the area of rectangles or right triangles through models, manipulatives or formulas [Topic 12]
  
• 
  calculate the area of polygons or irregular figures on grids [Topic 12]
  
• 
  convert the volume of rectangular prisms (cubes) using models, formulas or manipulatives [Topic 12]
  
• 
  apply all measures using appropriate units (benchmarks: inch to 1/8 inch; foot; centimeter to 0.5 centimeter; meter to 0.5 centimeter; yard; mile (use in scale questions); kilometer (use in scale questions); 12 inches in 1 foot; 100 centimeters in 1 meter; 3 feet in 1 yard; 36 inches in 1 yard; 10 millimeters in 1 centimeter; hour to 1 minute; day; year; 24 hours in 1 day; 7 days in 1 week; 365 days in 1 year; 60 seconds in 1 minute; 60 minutes in 1 hour; C degrees and F degrees to 1 degree; quart to 1 ounce; gallon; pint; 32 ounces in 1 quart; 4 quarts in 1 gallon; 2 pints in 1 quart; kilogram; gram to whole gram; pound to 1 ounce; 16 ounces in 1 pound; angle degrees to 2 degrees [Topic 14]
  
• 
  use units of measures appropriately and make conversions within systems when solving problems across the content strands[Topic 14]
  
• 
  use standard units in the customary and metric systems [Topic 14/Science Curriculum]
  
• 
  carry out simple unit conversions [Topic 14]
  
• 
  determine what happens to measurements of two-dimensional shapes such as its perimeter and area when the shape is changed [Topic 12]
  

• 
  use both metric and customary systems of measurement
  
• 
  convert from one unit to another within the same system
  
• 
  solve problems using metric or customary measures
  
• 
  find precise and estimated angle measures
  

• 
  use both metric and customary systems of measurement
  
• 
  convert from one unit to another within the same system (1-5, 3-6)
  
• 
  select and use units of appropriate size and type to measure angles, perimeter, area, surface area and volume
  

• 
  use both metric and customary systems of measurement
  
• 
  convert from one unit to another within the same system (4-2)
  
• 
  select and use units of appropriate size and type to measure angles, perimeter, area, surface area and volume
  

• 
  use non-standard tools to compare length and weight
  

• 
  measure with non-standard and standard tools
  
• 
  use appropriate tools to make comparisons and estimates
  

• 
  measure with multiple copies of the same size, (e.g., paper clips laid end to end)
  
• 
  use repetition of a single unit to measure something larger that the unit, (e.g., measuring the length of a room with a single meter stick)
  
• 
  use appropriate tools to measure length, weight, volume, area, and time
  
• 
  develop common referents for measures to make comparisons and estimates (body benchmarks)
  

• 
  estimate perimeters, area, and volumes of irregular shapes (benchmarks: quart to whole quart; kilogram to whole kilogram; gram to whole gram; pound to whole pound)
  
• 
  apply appropriate standard units and tools to measure length, area, volume, time, and temperature (benchmarks: hour to 5 minute interval; day; year; 24 hours in 1 day 7 days in 1 week; 365 days in 1 year; C degrees and F degrees to 1 degree)
  
• 
  use benchmarks to estimate measurements (benchmarks: inch to ½ inch; foot to whole inch; centimeter to whole centimeter; meter to whole centimeter)
  
• 
  use formulas to find the area of a rectangle
  
• 
  analyze the parts of rectangular solids
  

• 
  estimate perimeter, area and volume of irregular shapes
  
• 
  apply standard units and tools to measure length, area, volume, weight, time, temperature and the size of angles
  
• 
  use benchmarks to estimate measurements
  
• 
  use formulas to find area of rectangles and related triangles and parallelograms
  
• 
  develop strategies to determine the surface areas and volumes of rectangular solids
  

• 
  develop strategies for estimating perimeter, area, and volume of irregular shapes [Topic 12,14]
  
• 
  select appropriate units and tools to measure length, area, volume, weight, time, temperature, and the size of angles [Topic 12,14]
  
• 
  use the benchmarks to estimate measurements
  
• 
  use formulas to find the area of rectangles and related triangles and parallelograms [Topic 12]
  
• 
  determine the surface area and volume of rectangular solids [Topic 12]
  

• 
  use different techniques for estimating areas and perimeters of non geometric figures
  
• 
  determine area of triangles and quadrilaterals using models or formulas
  
• 
  identify the area of an object as the number of unit squares needed to cover it
  
• 
  determine perimeter of polygons using models or formulas (incorporating variables)
  
• 
  identify the perimeter of an object as the number of units of length needed to surround it
  
• 
  develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids and circles and develop strategies to find the area of more complex shapes
  
• 
  determine volume of rectangular prisms using models or formulas
  
• 
  solve problems involving perimeter, area, and volume using models or formulas
  
• 
  describe the relationship of radius to diameter and diameter to circumference
  
• 
  solve problems involving circle measures
  
• 
  express all measures using appropriate units (linear, square, or cubic)
  
• 
  measure and use units of measure (length, time, temperature, capacity, mass, weight, angles & rotation) appropriately and consistently
  

• 
  select and apply techniques and tools to accurately find surface area and volume (8-9, 8-10)
  
• 
  construct nets (flat patterns) of rectangular and triangular prisms, cylinders, and pyramids to explore the relationship between surface area of a box and the area of its faces (8-9)
  
• 
  recognize the number of cubes in the bottom layer of a box prism is equal to the area of the base (8-10)
  
• 
  develop strategies to determine the surface area and of selected prisms (rectangular and triangular), pyramids, and cylinders (8-9, 8-10)
  
• 
  develop strategies to determine the volume of selected prisms (rectangular and triangular), pyramids, and cylinders (8-9, 8-10)
  
• 
  solve problems involving scale factors, using ratio and proportion (5-2, 5-3, 5-4, 5-5, 5-6)
  
• 
  solve simple problems involving rates and derived measurements for such attributes as velocity and density
  

• 
  use common benchmarks to select appropriate methods for estimating measurements
  
• 
  solve simple problems involving growth and rates
  
• 
  select and apply techniques to accurately measure the surface area and volume of rectangular prisms, triangular prisms, cylinders, pyramids, or cones (8-3, 8-4, 8-5, 8-6, 8-7)
  
• 
  apply concepts of similarity to determine the length of sides of similar triangles (4-4)
  

• 
  collect data about themselves – answer questions asked about themselves
  
• 
  sort items into groups
  
• 
  use picture graphs or objects to represent items in the groups
  

• 
  develop simple questions to collect data about themselves
  
• 
  sort the information and organize it into groups
  
• 
  use tally marks, objects, pictographs, bar graphs and pictures to represent data
  

• 
  pose questions and gather data about themselves and their surroundings
  
• 
  sort and classify objects according to their attributes and organize data about the objects
  
• 
  represent data using objects, pictures and graphs
  

• 
  interpret a given representation such as line plots, tally charts, tables, line graphs, or bar graphs to answer questions related to the data
  
• 
  collect data using observations, surveys and experiments
  
• 
  recognize the differences in representing categorical and numerical data
  
• 
  formulate a question and develop a plan to collect information to address the question
  
• 
  answer questions related to the data
  

• 
  interpret a given representation using line plots, tables, bar graphs, pictographs, or circle graphs to answer questions related to the data, to analyze the data, to formulate or justify conclusions, to make predictions, or to solve problems
  
• 
  recognize the differences in representing categorical and numerical data
  
• 
  design investigations to address a question and consider how data-collection methods affect the nature of the data set
  

• 
  interpret a given representation using tables, bar graphs, circle graphs, or line graphs to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems [Topic 18/Science Curriculum]
  
• 
  collect data using observations, surveys, and experiments [Topic 18/Science Curriculum]
  
• 
  recognize the differences in representing categorical and numerical data [Topic 18/Science Curriculum]
  

• 
  formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population and show findings in an appropriate display
  
• 
  use bar graphs, tables, line graphs, and stem-and-leaf plots to display data
  

• 
  create and interpret circle graphs, scatter plots, and histograms (11-1, 11-2)
  
• 
  analyze data to formulate and justify conclusions
  
• 
  make predictions and solve problems based on a set of data
  
• 
  use tables, line graphs, scatter plots, circle graphs, and stem and leaf plots to organize and display data (11-1, 11-2, 11-3)
  
• 
  distinguish between discrete and continuous data ( Teachers must provide examples that will help students recognize the significance of these different representations.)
  
• 
  interpret and analyze linear relationships represented on a graph (9-6)
  
• 
  formulate questions and design a study that can be answered through the collection of data (11-7)
  
• 
  identify or describe the best representation to display data or a situation (11-3, 11-4)
  

• 
  identify the representation that best displays a given set of data or situation (9-9)
  
• 
  formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population (While studying graphic representations the teacher should have students use their knowledge to conduct and experiment or study.)
  
• 
  create and use histograms, box-and-whisker plots, line graphs, and scatter plots (9-2, 9-5, 9-6, 9-7)
  

• 
  begin to make observations and comparisons between sets on a representation (i.e., picture graph, bar graph, objects)
  

• 
  use mathematical language to interpret data from various representations (i.e., picture graph, bar graph, table, chart)
  
• 
  compare data (greater than, less than, same amount) within the same representation (i.e., picture graph, bar graph, table or chart)
  

• 
  use mathematical language to describe parts of the data and the set of data as a whole to determine what the data show
  

• 
  analyze patterns, trends or distributions in data in a variety of contexts by determining or using most frequent (mode), least frequent, largest, or smallest
  
• 
  identify or describe representations or elements of representations that best display a given set of data or situation using line plots, tally charts, tables or bar graphs
  
• 
  compare different representations of the same data and evaluate how well each representation shows important aspects of the data
  

• 
  analyze patterns, trends, or distributions in data in a variety of contexts by determining or using measures of central tendency (median or mode), or range
  
• 
  describe the shape and important features of a set of data and compare related data sets with an emphasis on how the data are distributed
  
• 
  use counting techniques to solve problems involving combinations or simple permutations (i.e., given a map—determine the number of paths from point A to point B) using a variety of strategies (i.e., organized lists, tables, tree diagrams, or others)
  
• 
  design investigations to address a question and consider how data-collection methods affect the nature of the data set
  

• 
  analyze patterns, trends, or distributions in data in a variety of contexts by determining or using measures of central tendency (mean, median, or mode) or range to analyze situations, or to solve problems [Topic 18/Science Curriculum]
  
• 
  identify or describe representations or elements of representations that best display a given set of data, or situation, consistent with the representations required in tables, bar graphs, circle graphs or line graphs Topic 18/Science Curriculum]
  
• 
  describe the shape and important features of a set of data and compare related data sets, with an emphasis on how the data are distributed [Topic 18/Science Curriculum]
  

• 
  compare sets of data using median and range
  
• 
  use mean, median, and mode as ways to describe what is typical about a set of data
  
• 
  use mean, median, mode and range to analyze situations and solve problems
  
• 
  determine the effect of range and outliers on mean, median, mode
  

• 
  analyze patterns, trends, or distributions of data to determine the effect on mean, median, mode (1-10)
  
• 
  use mean, median, mode and range to analyze situations and solve problems (1-10)
  
• 
  determine the effect of outliers on mean, median, mode (1-10)
  
• 
  evaluate a data sample to determine bias (11-4)
  

• 
  compare data using mean, median, range, percentiles, and data displays (Chapter 9)
  
• 
  compare data using tables, stem-and-leaf plots, histograms, and box-and-whisker plots (Chapter 9)
  
• 
  describe the correspondence between data sets and their graphical representations (line graphs, scatter plots, histograms, box-and-whisker plots) (Chapter 9)
  
• 
  analyze data using patterns and trends of central tendency (mean, median, mode, dispersion (range), outliers, quartile values, lines of best fit) (Chapter 9)
  
• 
  analyze data displays to formulate or justify conclusions, make predictions or solve problems (Chapter 9)
  

N/A

N/A

• 
  discuss events related to students’ experiences as likely or unlikely
  

• 
  make and justify predictions that are based on data
  
• 
  analyze data to formulate conclusions or to make predictions
  

• 
  propose and justify conclusions and predictions that are based on data and design studies to further investigate the conclusions or predictions
  
• 
  ask new questions and makes connections to real world situations
  
• 
  decide the most effective method (e.g., survey, observation, experimentation) to collect data (numerical or categorical) necessary to answer questions
  
• 
  collect, organize, display, and analyze the data to draw conclusions about the question or hypothesis being tested, and make appropriate predictions
  

• 
  propose and justify conclusions and predictions that are based on data and design studies to further investigate the conclusions or predictions [Topic 18/Science Curriculum]
  
• 
  organize data using tables, bar graphs, or line graphs to answer questions related to the data; or to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems [Topic 18/Science Curriculum]
  

• 
  interpret representations, including circle graphs, line graphs, and stem and leaf plots to justify conclusions, make predictions/inferences, or solve problems
  
• 
  decide the most effective method (e.g., survey, observation, experimentation) in response to a teacher or student generated question or hypothesis to:
  

• 
  collect the data (numerical or categorical) necessary to answer the question
  
• 
  collects, organizes, and appropriately displays the data
  
• 
  analyzes the data to draw conclusions about the question or hypothesis being tested
  
• 
  when appropriate makes predictions
  
• 
  asks new questions
  
• 
  makes connections to real world situations
  

• 
  use observations to make conjectures based on the differences between two or more samples (Chapter 11)
  
• 
  analyze data and justify interpretations based on the data (Chapter 11)
  
• 
  analyze data to draw conclusions or make predictions (Chapter 11)
  

• 
  use information drawn from samples to make conclusions about populations (Chapter 9)
  
• 
  make conjectures about possible relationships between two characteristics of a sample on the basis of scatter plots of the data and approximate lines of fit (Chapter 9)
  
• 
  use conjectures to formulate new questions and plan new studies to answer them ( Chapter 9)
  
• 
  decide the most effective way to investigate a teacher- or student-generated question or hypotheses then carry out the investigation (Chapter 9)
  
• 
  analyze data, make predictions, ask new questions, and connect the analysis to real world situations (Chapter 9)
  

N/A

N/A

N/A

• 
  determine the likelihood of the occurrence of an event (using “more likely,” “less likely,” or “equally likely”)
  
• 
  gather data, predict the probability and draw simple conclusions
  

• 
  find all possible combinations and arrangements within certain constraints involving a limited number of variables
  
• 
  predict the likelihood of an event using “more likely,” “less likely,” “equally likely,” certain, or impossible and test the prediction through experiments
  
• 
  determine if a game is fair
  
• 
  collect, organize, and appropriately display the data
  

• 
  show a probability event in which the sample space may or may not contain equal outcomes and determine the theoretical probability of an event and express the result as part to whole (e.g., two out of five)
  
• 
  describe events as likely or unlikely and discuss the degree of likelihood using such words as certain, equally, likely and impossible
  
• 
  predict the probability of outcomes of simple experiments and test the prediction
  

• 
  identify a probability event in which the sample space may or may not contain equally likely outcomes [Topic 20]
  
• 
  determine the experimental or theoretical probability of an event and express the result as a fraction[ Topic 20, Topic 9]
  
• 
  predict the likelihood of an event as a fraction and test the prediction through experiments; and determine if a game is fair[Topic 20]
  
• 
  describe events as likely and unlikely and discuss the degree of likelihood using words such as certain, equally, likely, and impossible [Topic 20]
  
• 
  predict the probability of outcomes of simple experiments and test the predictions [Topic 20]
  
• 
  obtain probabilities of the likelihood of an event with the representation of a number such as from 0-1 [Topic 20]
  
• 
  select the most effective method (e.g. survey, observation, experiment) to collect data (numerical or categorical) necessary to answer the questions or hypothesis being tested [Topic 18]
  
• 
  collect, organize and appropriately display data, then analyze it to draw conclusions about the question or hypothesis [Topic 18]
  
• 
  make predictions about a question or hypothesis and ask new questions and make connections to real world situations [Topic 20]
  

• 
  use organized lists, tables, tree diagrams, models and the Fundamental Counting Principle to compute probabilities for simple compound events
  
• 
  explain the difference between theoretical and experimental probability
  
• 
  predict the theoretical probability of an event and tests the prediction through experiments/simulations and designs fair games
  
• 
  compute the theoretical and experimental probability of an event in a problem solving situation
  

• 
  identify the differences between finding probability and odds (12-1)
  
• 
  compare and contrast experimental and theoretical probabilities (12-1, 12-2)
  
• 
  determine experimental or theoretical probabilities of events in problem solving situations (12-1, 12-2)
  
• 
  predict the theoretical probability of an event then test the prediction through experiments and simulations
  
• 
  use probability with proportionality to make predictions
  
• 
  use the following counting techniques to solve problems involving combinations and permutations: organized lists, tables, tree diagrams, area models, and the Fundamental counting Principle (12-3, 12-4) (The teacher must take time to make sure students understand how to use an area model as a counting technique.)
  

• 
  determine experimental or theoretical probabilities of events in problem solving situations then compare and contrast these probabilities (10-1)
  
• 
  predict the theoretical probability of an event then test the prediction through experiments and simulations (10-2)
  
• 
  use the following counting techniques to solve problems involving combinations and permutations: organized lists, tables, tree diagrams, area models, and the Fundamental counting Principle (12-3, 12-4) (The teacher must take time to make sure students understand how to use an area model as a counting technique.)
  

Students will demonstrate the ability to:

• 
  Compare two numbers with and without a number line, including:
  
  • 
    whole numbers
    
  • 
    integers
    
  • 
    rational numbers
    
  • 
    irrational numbers
    
• 
  Represent the same number in a variety of ways; including:
  
  • 
    fractions
    
  • 
    decimals
    
  • 
    percents
    
• 
  Apply the properties of the real number system; including:
  
  • 
    associative
    
  • 
    commutative
    
  • 
    distributive
    
  • 
    identities and inverses
    
• 
  Factor numbers using primes and composites; including:
  
  • 
    prime factorization
    
  • 
    exponential representation
    

Students will demonstrate the ability to:

• 
  Combine like terms.
  
• 
  Apply the order of operations to simplify expressions
  
• 
  Compute powers and roots
  
• 
  Use basic operations to solve one variable equations and inequalities by isolating variables, including one-step, two-step, multi-step and equations with variables on both sides
  

• 
  Use basic operations to solve simple radical equations
  
• 
  Graph linear equations by plotting points
  
• 
  Recognize and use the direct variation, slope-intercept form, and standard form of a line for graphing
  
• 
  Solve and graph linear inequalities by:
  
  • 
    isolating the variable
    
  • 
    knowing the difference between an equation and an inequality
    
  • 
    graphing an inequality on a number line
    
  • 
    graphing a compound inequality on a number line
    
  • 
    graphing an inequality in the coordinate plane
    
  • 
    knowing that multiplying or dividing by a negative reverses the direction of the inequality
    

• 
  Solve systems of linear equations and inequalities by using the following methods:
  

• 
  graphing
  
• 
  substitution
  
• 
  elimination
  

• 
  Solve equations which contain rational expressions by:
  

• 
  identifying rational expressions
  
• 
  applying operations to solve rational equations
  
• 
  using the Euclidean Algorithm to simply rational expressions
  
• 
  analyzing problem situations involving percentages
  

• 
  Solve quadratic equations by factoring, which includes:
  

• 
  recognizing quadratic equations
  
• 
  finding the greatest monomial factor
  

• 
  factoring using reverse Foil or Division method
  
• 
  applying the zero product property
  

• 
  Use polynomials, including:
  

• 
  identifying, adding, and subtracting polynomials and their parts
  
• 
  identifying and factoring using a common monomial factor
  
• 
  multiplying and dividing polynomials
  

• 
  recognizing special binomials (square binomial, perfect squares, difference of squares)
  

• 
  Judge the reasonableness of answers
  
• 
  Use permutations and combinations as counting techniques
  

Students will demonstrate the ability to:

• 
  Add, subtract, multiply, and divide whole numbers, integers, and rational and irrational numbers using mental or paper-and-pencil calculations for simple cases and technology for complicated cases
  
  • 
    Calculate benchmark perfect squares and related roots
    
  • 
    Determine or estimate the part of a number using percents and related fractions
    
• 
  Judge the reasonableness of solutions
  

Students will demonstrate the ability to:

• 
  Define, compare, and contrast the ideas of relation and function
  
• 
  Determine the difference between the independent and dependent variables
  
• 
  Analyze functions of two variables by investigating rates of change, intercepts, and zeros
  
• 
  Compare the properties of classes of functions, including linear, quadratic, and exponential functions
  

Students will demonstrate the ability to:

• 
  Explain the meaning of equivalent forms of expressions, equations, inequalities, and relations, write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—using mental or paper-and-pencil calculations for simple cases and technology for complicated cases
  
• 
  Translate words into algebraic symbols and equations
  
• 
  Use symbolic algebra to represent and explain mathematical relationships
  
• 
  Judge the reasonableness of solutions
  

Students will demonstrate the ability to:

• 
  Model and solve word problems using various representations (graphs, tables, equations) and determine the class of functions that best models the relationship
  
• 
  Draw reasonable conclusions about a situation being modeled
  

Students will demonstrate the ability to:

• 
  Approximate, interpret, and calculate rates of change from graphical and numerical data
  

Students will demonstrate the ability to:

• 
  Graph linear, quadratic, and exponential functions using graphing technology
  
• 
  Determine best fit lines for a set of data
  
• 
  Graph scatter plots using graphing technology
  
• 
  Use word processing and spreadsheet tools to communicate solutions to complex problems
  
  • 
    Analyze intercepts, domain, range, maximum and minimum values
    
  • 
    Analyze rates of change
    

Students will demonstrate the ability to:

• 
  Use Cartesian coordinates to analyze algebraic situations
  

Students will demonstrate the ability to:

• 
  Make decisions about units and scales that are appropriate for problem situations involving measurement
  

Students will demonstrate the ability to:

• 
  Solve complex problems involving ratios and proportions
  
• 
  Use the factor label method to assist in computation
  

Students will demonstrate the ability to:

• 
  Understand scatter plots and use them to display data
  
• 
  Determine measures of central tendencies for a set of data